John Baez

Higher Gauge Theory

Gauge theory describes the parallel transport of point particles using the
formalism of connections on bundles. In both string theory and loop quantum
gravity, point particles are replaced by 1-dimensional extended objects:
paths or loops in space. This suggests that we seek some sort of "higher
gauge theory" that describes parallel transport as we move a path through
space, tracing out a surface. To find the right mathematical language for
this, we must "categorify" concepts from topology and geometry, replacing
Lie groups by Lie 2-groups, bundles by 2-bundles, and so on. Some interesting
examples of these concepts show up in the mathematics of topological quantum
field theory, string theory and 11-dimensional supergravity.